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University Microfilms, a Xerqxcompany, Ann Arbor, Michigan 70 - 19,311 DUREN, Lowell Reid, 1940- AN ADAPTATION OF THE MOORE METHOD TO THE TEACHING OF UNDERGRADUATE REAL ANALYSIS— A CASE STUDY REPORT. The Ohio State University, Ph.D., 1970 Education, theory and practice University Microfilms, A XERQXCompany, Ann Arbor, Michigan © Copyright by Lowe 11 Re i d Du ren 1970 THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED AN ADAPTATION OF THE MOORE METHOD TO THE TEACHING OF UNDERGRADUATE REAL ANALYSIS— A CASE STUDY REPORT DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Lowe 11 Reid Duren, B.S., M.N.S. ****** The Ohio State University 1970 Approved by Adviser College of Education ACKNOWLEDGMENTS I wish to express my thanks to my adviser, Dr. Harold Trimbl Successful graduate study at Ohio State would have been impossible for me without his patience and encouragement. The completion of this dissertation owes much to Dr. Trimble's assistance and confi­ dence in me. Thanks are also due to the other members of my reading committee, Dr. Herbert Coon and Dr. Alan Osborne. In a large, some­ times impersonal, university, it is reassuring to find such warm and human gentlemen. I am indeed grateful to my colleagues at Western Maryland college, Professors Spicer, Sorkin, Lightner, Jordy, and Eshleman, for their conversations and assistance in gathering information relevant to this study, and their moral support during completion of it . I especially want to thank my departmental chairman, Dr. James Lightner, for his many valuable comments, assistance, and encouragement. Special recognition is also due the students: Sue Robertson Randy Klinger, Robert Gagnon, Kevin Fried, Bill E llio tt, Owen Ecker, Ray Brown, and Dave Baugh. Finally, I wish to thank my wife, Reba, for putting up with me through years of study and the completion of this dissertation. VITA September 22, 19A O ............................ Born, Wynnewood, Oklahoma 1962 ............................................................ B.S., Southwestern State College Weatherford, Oklahoma 1962-1963 .............................................. Teacher, Sayre Junior College Sayre, Oklahoma 1963-1965 .............................................. Teacher, Moore High School Moore, Oklahoma 1965 ............................................................ M.N.S., University of Oklahoma Norman, Oklahoma 1965-1966 .............................................. National Science Foundation Academic Year Institute, The Ohio State University, Columbus, Ohio 1966-1968 .............................................. Assistant Professor, Department of Mathematics, Wisconsin State University, Whitewater, Wisconsin 1968- ....................................................... Assistant Professor, Department of Mathematics, Western Maryland College, Westminster, Maryland FIELDS OF STUDY Major Field: Mathematics Education Studies in Mathematics Education. Professors Harold C. Trimble and Alan R. Osborne Studies in Teacher Education. Professor Herbert L. Coon i i i TABLE OF CONTENTS ACKNOWLEDGMENTS VITA LIST OF TABLES v LIST OF ILLUSTRATIONS vi Chapter INTRODUCTION Defini tions More on Moore Some Questions Origin and Background of the Problem Statement of the Problem Review of Related Literature Design of the Study The course The students Data collected Course content Organization of the Dissertation 11 THE COURSE 23 Progress Through the Course Introductory material Section five Section six Picture "proofs" The remaining material The hour test Daily progress Conduct of the Classes Reactions of the Instructor The classes The interviews Summary iv I I I . STUDENT PERFORMANCE *f3 Student Number One Backg round Performance Student Number Two Background Performance Student Number Three Background Performance Student Number Four Background Performance Student Number Five Background Performance Student Number Six Background Performance Student Number Seven Background Performance Student Number Eight Background Performance Summary IV. EXAMINATION OF QUESTIONS.................................................................... 78 The Questions Summary V. INFORMAL CONCLUSIONS .............................................................................. 85 Limitations and Disclaimers Data sources Influence of grades Influence of the study Influence of friendship The informal nature Specificity of the class Comparison with lecture Some Conclusions Opinions and Questions The role of the instructor The personality of the instructor Proof techniques Independent study and learning Concluding Remarks v APPENDIX A........................................................................* .................................................................100 B.............................................................................................................................................118 C.............................................................................................................................................125 D.................................................. 129 E.............................................................................................................................................135 F . ............................................................................................................. 144 G. , ...................................................................................................................................156 BIBLIOGRAPHY ................................................................................................................... 19* v I LIST OF TABLES Table Page 1. Class P r o f ile .................................................................................................. 18 2. Dai ly P ro g re s s.............................................................................................. 33 3. Work and Achievement ................................................................................ ^6 vi i LIST OF ILLUSTRATIONS Figure Page 1. Picture Proof ..................................................................................................... 29 2. An Exception .................................................................................................... 30 vi i i CHAPTER I INTRODUCTION It is commonly accepted that for a student to learn mathe­ matics he must do mathematics. The ways in which students are led to do mathematics are quite varied and there is no unique method. In fact, the methods used to teach collegiate mathematics are prob­ ably as diverse as the subject matter its e lf and the people teaching it. This is not necessarily bad, for as Polya states: ". there are as many good methods as there are good teachers." [35: 114].^ Still it seems that mathematics teachers, in spite of their different teaching methods, have a common goal. That is, they want to get the students actively involved in doing mathematics. Moise states this well when he says: We can "cover" very impressive material, if we are willing to turn the student into a spectator. But if you cast the student in a passive role, then saying that he has "studied" your course may mean no more than saying of a cat that he has looked at a king. Mathematics is something that one does. [29: *»09l • Teaching can be defined as a dialogue between the teacher and the student. By considering forms this dialogue takes, one is able to examine how the student is led to do mathematics. For exam­ ple, two popular methods of teaching collegiate mathematics are the Whe firs t set of digits represents the number of the biblio­ graphic entry; the second represents the page number. "lecture method" and the "Moore method."' The choice of these two methods as examples was not motivated by a desire to show one method superior to the other, and no such claim will be made in this disser­ tation. Rather, these examples were chosen because the lecture is so commonly used and it contrasts so well with the Moore method. Further more, this dissertation is based on an adaptation of the Moore method. Before proceeding with this discussion, it might be well to consider more precisely some of the terms being used. Defini tions Broudy defines teaching method to be: . the formal structure of the sequence of acts commonly denoted by instruction. The term covers both the strategy and '"Moore method" refers to the teaching style of the mathema­ tician Robert Lee. Moore. (The teaching style is described in later sections of this chapter.) R. L. Moore was born in Dallas, Texas, on November 14, 1882. He received B.S. and A.M. degrees from the University of Texas in 1901, and his Ph.D. from the University of Chicago in 1905. His doctoral research in the field of geometry was directed by Professor E. H. Moore. Further research by R. L. Moore in his speciality, plane analysis situs, contributed to the creation of a new area of mathematics now known as point set topology. After teaching at the University of Tennessee, Princeton University, Northwestern University, and the University of Pennsylvania, Professor Moore returned to the University of Texas in 1920 and has remained there since. He served as distinguished professor from 1937 to 1953, and his teaching career continued through the spring of 1969* In addition to teaching and his own research, Professor Moore has directed the Ph.D. research of a vast number of students. Many of these students made very significant contributions to the area of point set topology
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